I have heard it said, though I have not read up on this, that at age 12 the minor child may express a preference as to which parent he/she would prefer to live with and that this preference is to be granted under law.

My ex is trying to tell me that our daughter has the right to determine terms of visitation at age 12, i.e. "I don't want to see Daddy any more." My ex is throwing this up in my face now and I think she is confused as to custody versus visitation. I feel reasonably sure that it is the matter of custody that the minor child may have a say in at age 12. Anybody ever dealt with this?

If visitation is under court order, only the court can alter the terms. I imagine judge can decide what input to consider, including how much weight to give to the child's preference. The judge's decision will likely be focused on what's best for the child, which is usually fostering a good relationship with both parents (unless there's a concrete reason why contact with one parent would not be good for the kid).

In practice, you and your ex can practice whatever visitation schedule you want. As long as you both agree and nobody complains to the court they're not going to monitor to ensure you're adhering to the terms in the court order. But even then if you want to agree on different terms it's best to put in a stipulation and order to the court and make it official in case somebody decides later that they're upset.